Advances in Industrial Biotechnology Category: Biotechnology Type: Short Communication

# The AT -{Transmuted - X} Family of Distributions

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Accepted Date: Mar 07, 2019
Published Date: Mar 18, 2019

## Abstract

The Exponential Transmuted Exponential distribution (ETE) appeared in and in this paper we present a new generalization of the ETE distribution based on an application of the Ampadu-G family of distributions that appeared in [1,2]. We also show the new distribution is a good fit to some real-life data, indicating practical significance. As a further development, we propose a new class of distributions based on the structure of the weight function introduced in [3].

## Keywords

AT - X(W) family of distributions; Exponential Distribution; Transmuted Distribution

## PRELIMINARIES

At first we recall the following definitions

#### Definition 1.1.

Let λ > 0, ξ > 0 be a parameter vector all of whose entries are positive, and x ? R [2]. A random variable X will be said to follow the Ampadu-G family of distributions if the CDF is given by

And the PDF is given by

Where the baseline distribution has CDF G(x; ξ) and PDF g(x; ξ)

#### Definition 1.2.

Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ) [2]. We say a random variable S is AT - X(W) distributed of type II if the CDF can be expressed as either one of the following integrals

or

Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)

## APPLICATION

As an application we consider the data on patients with breast cancer [1]. We assume the random variable T is exponentially distributed. So that the CDF of T is given by

G(t; d) = 1 - e-dt
For t, d > 0 and the PDF is given by

g(t; d) = de-dt

For t, d > 0. Now we recall the following, for some baseline distribution K(x), the transmuted family of distributions has CDF given by

(1 + b)K(x) - bK2(x)

Where b ? [-1, 1]. Now we assume X is transmuted exponentially distributed, so that if

K(x):= 1 - e-ax
For x, a > 0, then the CDF of X is given by

F(x; a, b) = (b + 1) (1 - e-ax) - b (1 - e-ax)2
And the PDF is given by

f(x; a, b) = a(b + 1) e-ax - 2abe-ax (1 - e-ax)

Now from the first integral in 3.2 Definition, we have the following

#### Theorem 2.1.

The CDF of the A{Exponential}-{Transmuted Exponential} distribution is given by

Where x, a, c, d > 0 and b ? [-1, 1]

#### Remark 2.2.

If a random variable Q has CDF given by the above theorem, write

Q ~ AETE(a, b, c, d)

The PDF of the AETE distribution can be obtained by differentiating the CDF
The AETE distribution is seen to be a good fit to real life data as shown in the figure 1.

Figure 1: The CDF of AETE(0.0133756, 0.216229, 3.93753, 0.910041) fitted to the empirical distribution of the data on patients with breast cancer [1].

## FURTHER DEVELOPMENTS

Inspired by the structure of the weight function introduced in, we ask the reader to investigate some properties and applications of a so-called A{New T} - X(W) family of distributions of type II [3]. We leave the reader with the following.

#### Definition 3.1.

Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ). We say a random variable SNew is A{New T} - X(W) distributed of type II if the CDF can be expressed as the following integral.

Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)

## CONCLUDING REMARKS

Our hope is that the new class of distributions presented in this paper will find application in cancer modeling and forecasting

## REFERENCES

Citation: Ampadu CB (2019) The AT - {Transmuted - X} Family of Distributions. Adv Ind Biotechnol 2: 006.

Copyright: © 2019  Clement Boateng Ampadu, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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##### Industrial Biotechnology Biomaterials Bioprocessing Biomanufacturing Food Biotechnology

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